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<title>Simulations for Statistical and Thermal Physics</title>

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<h3 style="text-align:center;">Planar model</h3>

<p class="header_title">Introduction</p>

<p>A Monte Carlo simulation of the planar or XY model. The spins are located on
a square lattice and can point in any direction in the plane. The interaction
is given by -J <b>s</b><sub>1</sub> <sup><b>.</b></sup> <b>s</b><sub>2</sub>, where J is the energy of the interaction, <b>s</b><sub>1</sub> and <b>s</b><sub>2</sub> are 
two unit spin vectors on neighboring lattice sites.</p>
<center>
<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.xymodel.XYAnimationApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
</center>

<p class="header_title">Algorithm</p>

<p>The Metropolis algorithm is used and can be summarized as follows</p>

<ol>

<li>A spin is chosen at random and rotated by a random amount up to
a maximum value &#948;.</li>

<li>The change in energy &#916;E for the trial rotation is computed.</li>

<li>If &#916;E &#8804; 0, accept the flip. Else if exp(-&#916;E/T) &gt; r, where r is a uniform random number between 0 and 1, the flip is also accepted; otherwise reject the flip.</li>

<li>Repeat steps 2 and 3 for many 
Monte Carlo steps per spin (mcs).</li>

<li>Accumulate data for the various thermodynamic quantities</li> 
</ol>

<p class="header_title">Problems</p>

<ol>

<li>Run the simulation with the default parameters and observe the locations of the vortices using the default parameters. Follow the arrows as
they turn around a vortex. A vortex is indicated by a square box. What is the difference between a 
positive (blue) and negative (red) vortex? Does a vortex ever appear isolated?
Count the number of positive vortices and negative vortices. Is the number the same at all times?</li>

<li>Click the <tt>Reset</tt> button and change the temperature to 0.2. Make sure the initial configuration is 
set to "random." You should see quenched-in vortices which don't change with time. Are there an equal 
number of positive and negative vortices? Are there isolated vortices whose centers are more
than a lattice spacing apart?</li>

<li>Click the <tt>Reset</tt> button and set the initial configuration to "ordered" and the temperature
to 0.2. Also set steps per display to 100 so that the simulation will run much faster.
Run the simulation for at least 1000 mcs and record the thermodynamic data. Repeat for temperatures
from 0.3 to 1.5 in steps of 0.1. Plot the energy and specific heat versus the temperature. Near 
the specific heat and susceptibility peaks take more data in temperature intervals of 0.02. Do
the peaks in the susceptibility and specific heat occur at the same temperature? 
Is the vorticity (the mean number density of vortices) a smooth function of the temperature?</li>

<li>Near the peak in the susceptibility look at configurations showing the vortices. At the peak in susceptibility (the Kosterlitz-Thouless transition) is there any evidence
that the positive
vortices are moving away from the negative vortices? The Kosterlitz-Thouless transition is due to this unbinding
of vortex pairs.</li>

</ol> 

<p class="header_title">Java Classes</p>

<ul>
<li>XYAnimationApp</li>
<li>XYModel</li>

</ul>

<p class = "small">Updated 27 February 2007.</p>
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